Direct Evidence of Drift‐Compressional Wave Generation in the Earth's Magnetosphere Detected by Arase

We present the first direct evidence of an in situ excitation of drift‐compressional waves driven by drift resonance with ring current protons in the magnetosphere. Compressional Pc4–5 waves with frequencies of 4–12 mHz were observed by the Arase satellite near the magnetic equator at L ∼ 6 in the evening sector on 19 November 2018. Estimated azimuthal wave numbers (m) ranged from −100 to −130. The observed frequency was consistent with that calculated using the drift‐compressional mode theory, whereas the plasma anisotropy was too small to excite the drift‐mirror mode. We discovered that the energy source of the wave was a drift resonance instability, which was generated by the negative radial gradient in a proton phase space density at 20–25 keV. This proton distribution is attributed to a temporal variation of the electric field, which formed the observed multiple‐nose structures of ring current protons.

DCM is excited via resonant interactions with particles energized through the "bump-on-tail" distributions or gradient instability (Crabtree et al., 2003;Kostarev & Mager, 2017;Mager et al., 2013) or via coupling with the shear Alfvén mode (Mager et al., 2015;Mager & Klimushkin, 2017).Although a few theoretical studies have investigated the excitation of DCM, the most crucial excitation mechanism in a realistic situation remains to be elucidated, owing to a lack of in situ observations of the plasma distribution function in the magnetosphere.
This study derived a unique perspective on compressional Pc5 waves using observational data provided by the Arase satellite.Both the oscillation mode and generation mechanism of the waves were directly determined from the satellite data.Section 2 describes the data used in the study.Section 3 provides an overview of the wave properties and plasma environment.Section 4 presents a theoretical interpretation of the generation mechanism.Finally, Section 5 summarizes this study.

Data
In this study, we analyzed the data provided by the Arase satellite (Miyoshi, Shinohara, et al., 2018) and used 8-s spin-averaged magnetic field data (Matsuoka, Teramoto, Nomura, et al., 2018).The magnetic field vectors (B) were rotated in the mean field-aligned coordinate system.The parallel direction (∥) was determined from the moving magnetic field averaged over 10 min, which is much larger than the wave period (<250 s).The azimuthal (a) and radial (r) directions are positive eastward and outward, respectively.We used the 1-min electron density data obtained from the upper hybrid resonance frequency (Kumamoto et al., 2018) and ion flux data in the energy range from 3.8 eV/q to 184.2 keV/q, which were obtained using medium-energy particle experiment-ion (MEP-i; Yokota et al., 2017) and low-energy particle experiment-ion (LEP-i; Asamura, Kazama, et al., 2018;Asamura, Miyoshi, & Shinohara, 2018) mass analyzers.The time resolution of the mass analyzer was set to 8 s during the interval of interest.We calculated proton thermal pressure by combining the MEP-i measurements in the range 9.6-184.2keV/q with LEP-i measurements between 64 eV/q and 6.1 keV/q, following the approach used by Menz et al. (2017) and Imajo et al. (2019).The OMNI Web database provided 1-min solar wind and interplanetary magnetic field (IMF) data.The World Data Center for Geomagnetism, Kyoto provided the SYM-H, AE, AU, and AL indices, which measure geomagnetic activity.

Observation
During 03:30-04:30 UT on 19 November 2018, we found large amplitude oscillations in the parallel magnetic field component (Figure 1c) in the evening sector (magnetic local time (MLT) ≈ 21 hr) near the geomagnetic equator and close to the apogee of Arase's orbit at L ≈ 6.2 (See Figures S1a and S1b in Supporting Information S1).The amplitude of the parallel component is the largest among the three components at any latitudes, which indicates the observed waves are compressional waves.The compressional waves were preceded by transverse waves with a frequency ( f ) of approximately 4 mHz during 02:45-03:30 UT (Figures 1a and 1d).Both waves had comparable amplitudes but different dynamic spectra.As shown in Figures 1d and 1f, the compressional wave spectrum exhibits broadband oscillations in the wave-frequency range of 4-12 mHz, while the radial and azimuthal oscillations have narrow spectral peaks along the calculated eigen frequencies of standing Alfvén waves (gray lines).Therefore, we believe that the compressional and transverse waves should be discussed separately.The excitation mechanism of the transverse waves will be reported in a separate paper.
The geomagnetic condition was slightly disturbed by a few southward turnings of the IMF around 04:00 UT on 19th November, increasing the AE index up to approximately 190 nT.The AU index variations were more significant than the AL index from 02:00 UT to 07:00 UT on 19th November.This is not a typical signature of substorm activity.The solar wind velocity remained nearly constant at 330 km/s, while the proton density exceeded 20 cm 3 .The SYM-H index fluctuated owing to magnetospheric compression resulting from an increase in solar wind dynamic pressure on 18 November; however, the index was approximately 0 nT during the wave observation.The solar wind parameters in this interval are shown in Figures S1c-S1g in Supporting Information S1.
Figures 1g and 1h indicate that a "nose structure" of ring current protons (e.g., Ejiri et al., 1980) exhibited multiple energy bands at approximately 5 and 15 keV during 02:00-03:00 UT.The proton perpendicular pressure P ⊥,H+ reached approximately 3 nPa, which is slightly higher than the proton pressure near the midnight sector during quiet time (Lui & Hamilton, 1992).The plasma β of the protons reached 1; it is expressed by the equation β ⊥,H+ = 2μ 0 P ⊥,H+ /B, where μ 0 and B are the vacuum permeability and magnetic field intensity,  (d-f) Wavelet power spectra of the radial, azimuthal, and parallel magnetic fields.Gray lines show eigenfrequencies of poloidal and toroidal standing Alfvén modes up to the seventh harmonic.The eigenfrequencies were calculated using the Tsyganenko (1989) model and the MHD wave equation developed by Singer et al. (1981).A power-law distribution of the plasma density and proton plasma are assumed in the calculation.(g, h) Proton omni-directional differential number fluxes measured by MEP-i and LEP-i mass analyzers, respectively.

Geophysical Research Letters
10.1029/2023GL107707 respectively.The B ∥ wave power and the plasma β peaked simultaneously around 03:50 UT.Subsequently, both parameters gradually decreased as the spacecraft moved to the apogee at approximately 04:20 UT and then left the equatorial region.The ion anisotropy parameter Γ was calculated by using the following equation: where P ⊥,s and P ∥,s are the perpendicular and parallel plasma pressures of the ion species s, respectively.Γ was below 0.5 at all times.The cold electron density near the apogee was approximately 150 cm 3 because of either plasmasphere expansion or duskside plasmaspheric bulge during quiet time, causing a low ion temperature (40-140 eV) at L > 5.The ion pressure, the plasma β, the ion anisotropy parameter, the cold electron density, and the ion temperature are shown in Figure S2 in Supporting Information S1.
We theoretically evaluated plasma pressure fluctuations using the perturbed distribution functions of DMM and DCM (δP ⊥,DMM and δP ⊥,DCM ), assuming a bi-Maxwellian distribution and a low wave frequency (Takahashi et al., 2022).
where δB is the perturbation of the magnetic field intensity.T ⊥ and T ∥ the perpendicular and parallel temperatures of the ion species s. δP ⊥ and δB were obtained from the differences between raw data and their 10-min moving averages.The radial distributions of the plasma pressure and the magnetic field were calucated from the radial motion of Arase on the outband orbit.Radial gradients of them at L = 6.1 were used for Equations 2 and 3.As evident from Figure 2a, DCM explains the observed pressure fluctuations (δP ⊥,Obs ) better than DMM.
The proton fluxes around the pitch angle α = 90 °are strongly modulated (Figure 2b).This pitch angle dependence is expected for wave-particle interactions with a compressional-mode wave confined around the magnetic equator.Figure 2c shows the residual proton fluxes of α = 90 °(δJ/J 0 , where J is the differential flux, δJ = J-J 0 , and J 0 is the 10-min moving average).The compressional wave modulated proton fluxes of energies in the range of 10-40 keV.These protons correspond to the higher-energy region of the outer portion of the nose structure (Figures 1g and 1h).We analyzed the residual fluxes using the Morlet wavelet to examine the energy dependence of the flux modulation.The dynamic spectra were averaged over 4-12 mHz and 03:30-04:30 UT. Figure 3a shows the averaged power spectral density of the residual fluxes, and Figure 3b shows the coherence between residual fluxes and δB ǁ with respect to energy.Both the power and coherence have maxima at 20-25 keV, which implies that the wave-particle interactions occurred in this energy range.
We calculated the azimuthal wave number m corresponding to the drift resonance of protons at 20-25 keV under the drift-bounce resonance condition (Southwood et al., 1969).
where ω is wave angular frequency, ω d and ω b are bounce-averaged drift angular velocity and the bounce angular frequency, respectively (Hamlin et al., 1961;Oimatsu et al., 2018), and K is an integer.As per the drift resonance condition, which is represented by the black curve labeled with K = 0 in Figure 3c, m values from 158 to 119 were obtained for the given energy range with high coherence.We also estimated m using the finite gyroradius effect (Su et al., 1977;Takahashi, Claudepierre, et al., 2018;Takahashi, Oimatsu, et al., 2018).In this analysis, we used proton fluxes having an energy of 19.1 and 25.5 keV from LEP-i measurements and 17.9 and 22.1 keV from MEP-i measurements at α = 90 °during 03:35-03:50 UT.The squares in Figure 3c show the estimated m values (blue and red squares indicate m values based on LEP-i and MEP-i measurements, respectively).The m values ranged from 130 to 104 with a standard deviation of approximately 50.This result is consistent with that obtained based on the drift resonance theory.The linear correlation between f and m based on the DCM dispersion relation (Rubtsov et al., 2018) predicted a wide range of m values because the compressional wave exhibited broadband spectra (4-12 mHz) during this event.

Interpretation
The observed compressional wave cannot be interpreted as a DMM wave because Γ is negative and δP ⊥,DMM is not consistent with δP ⊥,Obs .By contrast, the observations strongly suggest that the wave is a DCM wave.We calculated the DCM frequency via the gyrokinetic approach followed by Mager et al. (2013) for further validation.While the diamagnetic drift frequency provides an approximate value of DCM frequency (e.g., Takahashi et al., 2022), our analysis provides the first comparison between an observed frequency and DCM eigenfrequency calculated using the kinetic theory.We fitted a Maxwellian function to the observed proton distribution function to deduce the physical parameters describing the cold and hot populations of the protons.The first function is related to the main cold plasma population, whose peak was estimated to be located below the lower limit of the energy coverage of LEP-i.The second function describes hot proton population (>50 eV) whose peak is located at approximately 20 keV.These two populations were separated in the observations.Plasma pressure and plasma β are key parameters influencing the DCM frequency.Because the effect of cold protons on these parameters is insignificant, we considered only the hot proton population.We obtained the hot proton density N H+ and perpendicular temperature T ⊥ from the proton flux data for an energy range of 50 eV-180 keV.According to Mager et al. (2013), the DCM frequency is determined as follows: 2 3 where z = ω 1 /mω d and ω 1 is the DCM principal harmonic eigenfrequency. and are the diamagnetic angular velocities for protons at α = 90°due to temperature and density radial gradients, respectively.l b is the length of the particle path over a bounce period, Λ 1 = 0.5/R is the principal harmonic eigenvalue given by Equation 19in Mager et al. (2013), and R is the field line curvature radius.Z is the plasma dispersion function.The real part of ω 1 was obtained by substituting the proton temperature and density measured by Arase into Equations 6 and 7. Magnetic field curvature is the only prerequisite for the existence of a DCM wave, while either ∂T ⊥ /∂L > 0 and ∂N H+ /∂L < 0 must be satisfied or a "bump-on-tail" distribution must occur to trigger a drift-compressional instability (Crabtree & Chen, 2004;Crabtree et al., 2003).
Accurate calculation of the ∂T ⊥ /∂L and ∂N H+ /∂L values near the apogee is difficult; hence, we used two approximations: (a) radial gradients obtained from the radial movement of the spacecraft, and (b) zero constant gradients.As shown in Figure 4a, the observed f (white dots) is close to or lies between f 1 = ω 1 /2π values (black and magenta dots) calculated using the approximations.As z is inversely proportional to m, we estimated the timevarying m to obtain the temporal variation in f 1 during the observation.We used the finite gyroradius effect for several wave periods to determine m in the time domain.The finite flux data cadence and broadband characteristics of the compressional wave spectra introduce an error in the time-varying m; hence, a wide range of m values were obtained during observation.Owing to this error, f deviated by a few millihertz around the values indicated by the dots in Figure 4a.Figures 4b and 4c show the ∂T ⊥ /∂L and ∂N H+ /∂L values used for the first approximation.The values could not be obtained after 03:53 UT, when Arase approached the apogee.
∂T ⊥ /∂L may become overestimated since the spacecraft moved into the off-equator.This is because P ⊥,H+ decreases with increasing magnetic latitude for a given L-shell (Imajo et al., 2019).The uncertainty in ∂T ⊥ /∂L obtained from the spacecraft motion also increases.This may cause a considerable discrepancy between f and f 1 calculated with the first approximation (black dots) in the latter stages of the calculation.The dependence of ion fluxes on MLT and temporal variation can also cause an error in evaluating the radial gradient.
The radial ion temperature gradient during 03:00-03:30 UT is clearly positive, which might cause drift compressional instability (Figure S2d in Supporting Information S1), but the sign of the gradient around the spacecraft apogee is not stably positive (Figure 4b).Another possible mechanism for generating DCM waves is the instability caused by the drift resonance, which is a wave-particle interaction between drifting charged praticles and wave fields.While the drift resonance is considered as an excitation mechanism of the fundamental poloidal standing waves (e.g., Dai et al., 2013;Takahashi, Claudepierre, et al., 2018;Takahashi, Oimatsu, et al., 2018), some theoretical studies suggest that the gyrokinetic wave equation of DCM includes the effect of the drift resonance (e.g., Mager et al., 2013).The instability condition (Southwood et al., 1969) is expressed as follows.
where F, W, M res , q, B eq , and W res are the distribution function, particle energy, magnetic moment of resonance particles, elementary charge, magnitude of the magnetic field on the Earth's equatorial surface (∼29,400 nT), and resonance energy, respectively.To the best of our knowledge, no previous studies have examined the drift resonance instability condition as an excitation mechanism of DCM waves.
Figures 3e-3g show the results calculated at W res = 19.2keV and M res = 0.20 keV/nT.The first term in Equation 9 is always negative because the distribution function decreases with energy without the occurrence of any "bump-on-tail" signatures around the resonance energy (Figure 3e).As the spacecraft was close to the apogee, we used the proton flux data along the inward and outward guiding center directions to calculate ∂F/∂L (e.g., Yamamoto et al., 2018).For m < 0, the radial gradient of the distribution function (the second term in Equation 9) causes a drift resonance instability if ∂F/∂L < 0, which was satisfied several times during the observation (Figure 3f).Previous studies have shown that the gradient of the distribution function of ring-current ions can generate poloidal Alfvén waves as well (O.V. Mager, 2021;Mikhailova et al., 2022;Rubtsov et al., 2021;Yamamoto et al., 2019).For a general review of wave-particle interactions, refer to (Klimushkin et al., 2021).The periods corresponding to dF/dW > 0 roughly aligned with those corresponding to the occurrence of the wave packets (Figure 3g), indicating a strong correlation between the observed compressional wave and the destabilization condition of the drift resonance.The convective growth of waves or latitudinal inhomogeneity of resonance protons can result in a misalignment between wave amplification and the fulfillment of destabilization conditions.
The protons generating the waves comprise a nose structure with multiple energy bands (Figure 1h).The formation of the multiple-band nose structure may be associated with the complicated radial distribution of energetic protons.Ebihara et al. (2004) and Ferradas et al. (2016) showed that the lower-energy protons of the multipleband nose structure drift directly from the source location rapidly, whereas the higher-energy protons drift around the Earth.Both populations of protons reach the same location under a time-varying convection electric field.In this study, a proton population, which probably belonged to the higher-energy band, served as the energy source of the observed wave.When we considered 19.2 keV protons and traced back their drift motion at L = 6.2, we discovered that these protons launched at 20:12 UT on November 18 encircled the Earth and reached the spacecraft position when the radial gradient of the distribution function became negative (03:50 UT on November 19).Therefore, these protons were possibly injected during a short southward excursion of the IMF between 20:20 UT and 20:40 UT on November 18 (See Figure S1c in Supporting Information S1).Before the protons reached the spacecraft's position, the IMF remained southward continuously from 23:50 UT on November 18.In this interval, the open-closed separatrix of the drift path of these protons may shrink, releasing some protons trapped in the higher L-shells.In this case, the radial gradient of the distribution function is likely to turn negative, thereby generating waves through drift resonance.

Conclusions
Pc4-5 compressional ULF waves were observed by the Arase satellite near the magnetic equator in the evening sector (∼21 MLT) during slightly disturbed geomagnetic conditions.The observed waves had a broadband frequency spectrum of 4-12 mHz.Arase was located inside the plasmasphere (N e ∼ 150 cm 3 ); however, the plasma β reached approximately 1.The anisotropy parameter Γ was below 0.5, implying that the drift mirror instability cannot occur.The wave properties were consistent with those theoretically estimated by Mager et al. (2013) and Takahashi et al. (2022).The eigenfrequency of DCM, which was derived via the method followed by Mager et al. (2013), showed quantitative agreement with the observed wave frequency.The relationship

Geophysical Research Letters
10.1029/2023GL107707 between the magnetic field and proton pressure oscillations was confirmed using the theory proposed by Takahashi et al. (2022).These results led us to conclude that the observed wave was a DCM wave.
Coherent proton flux oscillations occurred simultaneously at 20-25 keV, suggesting wave-particle interactions between the compressional wave and ring current ions.As no "bumps" were observed in the proton distribution function, the compressional waves can be attributed to a positive radial gradient of ion temperature and drift resonance instability.The negative radial gradient of the distribution function at 19.2 keV was sufficiently high to cause instability.Assuming the resonance energy to be 20-25 keV, the azimuthal wave number m was found to be in the range of 160 to 120 using the drift resonance theory.These results are consistent with the estimate of m (∼ 130) derived from the finite gyroradius effect.
This study comprehensively analyzed the energy and radial gradients of the distribution function and was the first study to discover that the free energy of drift resonance is provided during DCM excitation.We suggest that the ring current ions are related to the nose structure as a source population of the resonating particles.The nose structure observed in this event had two earthward-extending energy bands at approximately 5 and 15 keV, suggesting that the source population was exposed to the temporal variations of the convection field in the magnetosphere.This may lead to the formation of an unstable spatial distribution; however, multipoint observations of the proton distribution function are required to validate the excitation scenario.Future multi-spacecraft missions in mesoscale physics are crucial for understanding the role of energetic ions in ULF wave excitation.

Figure 1 .
Figure 1.(a-c) Magnetic field oscillations in radial, azimuthal, and parallel components observed by the Arase satellite.For the parallel component, highpass filtered (>1.67 mHz) magnetic field is shown.(d-f)Wavelet power spectra of the radial, azimuthal, and parallel magnetic fields.Gray lines show eigenfrequencies of poloidal and toroidal standing Alfvén modes up to the seventh harmonic.The eigenfrequencies were calculated using theTsyganenko (1989) model and the MHD wave equation developed bySinger et al. (1981).A power-law distribution of the plasma density and proton plasma are assumed in the calculation.(g, h) Proton omni-directional differential number fluxes measured by MEP-i and LEP-i mass analyzers, respectively.

Figure 3 .
Figure 3. Energy-dependent Morlet wavelet spectra of (a) the power of residual flux oscillations and (b) the coherence between the residual flux and B ∥ averaged over 4-12 mHz and 03:30-04:30 UT on 19 November 2018.The red and blue lines show MEP-i and LEP-i fluxes, respectively.(c) Resonance energy calculated from the drift-bounce resonance theory as a function of m.The gray region indicates resonance energies at 20-25 keV.Red and blue squares denote the estimated values from MEP-i and LEP-i data, respectively.The horizontal bar on each square shows the standard deviation.(d) Band-pass filtered (4-12 mHz) B ∥ .1-min averaged gradients (e) ∂F/∂W, (f) ∂F/∂L, and (g) dF/dW of the proton distribution function at W res = 19.2keV and M res = 0.20 keV/nT.The gray region corresponds to dF/dW > 0.

Figure 4 .
Figure 4. DCM frequency calculation.(a) Wavelet amplitude function (WAF; Foster, 1996) of B ǁ is color-coded.White dots are wave frequencies extracted from the spectrum.Black dots denote the DCM eigenfrequency calculated using the first approximation (radial gradients) and the time-varying m obtained using the finite gyroradius effect.Magenta dots denote the DCM eigenfrequency calculated for the time-varying m and zero constant gradients.(b) Hot protons (>50 eV) perpendicular temperature gradient.(c) Hot protons density (red) and its gradient (black).